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This paper deals with the application of an anticipating and coordinating feedback control scheme in order to mitigate the long-term voltage instability of multi-area power systems. Each local area is uniquely controlled by a control agent (CA) selecting control values based on model predictive control (MPC) and is possibly operated by an independent transmission system operator (TSO). Each MPC-based CA only knows a detailed local hybrid system model of its own area, employing reduced-order quasi steady-state (QSS) hybrid models of its neighboring areas and even simpler PV models for remote areas, to anticipate (and then optimize) the future behavior of its own area. Moreover, the neighboring CAs agree on communicating their planned future control input sequence in order to coordinate their own control actions. The feasibility of the proposed method for real-time applications is explained, and some practical implementation issues are also discussed. The performance of the method, using time-domain simulation of the Nordic32 test system, is compared with the uncoordinated decentralized MPC (no information exchange among CAs), demonstrating the improved behavior achieved by combining anticipation and coordination. The robustness of the control scheme against modeling uncertainties is also illustrated.

The introduction of the competitive market into the electricity industry, due to the deregulation and liberalization of electrical power systems, has created economic incentives for power systems to operate closer and closer to their safety and stability limits [

The areas are typically interconnected through power transmission corridors that may carry heavy power flows in order to economically establish a common open electricity market and, technically, hopefully, to ensure a greater security margin, through the sharing of active/reactive power reserves. This interconnectedness, on the other hand, makes the large-scale power systems even more complex and harder to control by increasing the probability of creating more complicated loops that may destabilize the system, making it more vulnerable to global collapse, initiated by a local fault [

Voltage collapse, as a calamitous result of unmitigated local voltage instability, can often lead to a blackout in the system [

Our earlier work [

The DCMPC scheme enjoys the joint contribution of anticipation and coordination. In order to identify the sole contribution of coordination in the improved performance of the DCMPC and to justify that the added complexity (by requiring communication between areas) is indeed needed to stabilize the system, the DCMPC-based results are compared with an uncoordinated decentralized MPC approach;

The robustness of the DCMPC against modeling uncertainties is assessed. This is very important, as the DCMPC is a model-based scheme;

The DCMPC scheme complements (and does not replace) the existing secondary control layer of a generic voltage control hierarchy. Thus, the feasibility of practical implementation of the DCMPC in the current existing voltage control schemes is also briefly discussed.

The remainder of this paper is organized as follows. The need for anticipation and coordination of local control actions in multi-area power systems for an effective voltage control is illustrated in Section 2. A further elaboration of the DCMPC scheme is given in Section 3. Next, Section 4 presents simulation results with the uncoordinated decentralized MPC obtained from time-domain simulation of the realistic well-known Nordic32 test system. The robustness of the DCMPC is illustrated in Section 5. The practical implementation issues are discussed in Section 6. Possible extensions of the DCMPC are described in Section 7. Finally, conclusions are drawn in Section 8.

The anticipation and (more importantly) coordination of the effect of control actions in a network of many interacting components, in which each component is controlled by an independent CA, is a very challenging problem. The nonlinearity, especially arising from saturation, makes this even more important. Indeed, anticipation and coordination seems to be necessary in any network of interacting components in which a local perturbation can lead to global performance degradation. Some examples are:

voltage control in multi-area power systems;

control of traffic lights in an urban traffic network;

on-ramp metering to control freeway traffic, taking overflow into neighboring roads into account;

flood/irrigation control, where controllable gates can regulate the flow of water.

The DCMPC approach in this article is applied, as a case study, to the voltage control problem of electrical power systems. However, the DCMPC is generally applicable to many large networks of interacting dynamic components. It is useful whenever both anticipation (implemented via the model-based prediction of the effects of a local control action over a sufficiently long window of time) and coordination (avoiding the overall system being destabilized by unintended interactions between local control actions) are needed.

One of the most successful classes of closed-loop model-based anticipating schemes is the model predictive control (MPC) paradigm, also called receding/moving horizon control. MPC relies on an estimate of the current system states at discrete time instant _{k}_{c}_{k}_{k}_{+}_{H}_{k}_{k}_{+1}),…, _{k}_{+}_{N}_{−1})}, where _{k}_{k}_{k}_{k}_{+1}, each time predicting performance over a shifted window with the size of _{c}_{k}

Anticipation/prediction is a very useful means for designing an on-line, well-performing, long-term voltage controller. The anticipating voltage controller can anticipate, within a prediction window [_{k}_{k}_{+}_{H}

The model-based simulations for the local anticipating controllers must be fast enough in order to be applicable for real-time applications, such as on-line voltage control. We have developed, in [

The case study of voltage control in large-scale multi-area power systems is an interesting application for coordinated control, as the poorly coordinated (or uncoordinated) operation of the areas may increase the risk of blackouts. In other words, in highly-coupled multi-area power systems, the creation of destabilizing loops as a reaction to a local disturbance, due to the complicated interaction among CAs, can eventually lead to a voltage collapse. Thus, the control actions taken by (at least) neighboring CAs must be coordinated. It is noteworthy that the voltage in electrical power systems is, by nature, a “local” variable, unlike frequency, which is a “global” variable. This means that, in multi-area power systems, only electrically close areas interact for voltage control, and thus, there is no need to involve distant areas with negligible common interest in solving a local optimization problem. The latter makes possible a decomposition approach for voltage control, where the voltage control still remains a prerogative of the local utilities.

In the case of a simple and small interconnected power system, e.g., power systems consisting of, at most, two areas, or in the conventional radial distribution networks with the integration of distributed generation units, the voltage coordination may be obtained by heuristic

Designing a possible coordination strategy is a major challenge involving many issues, such as what information to communicate, how to avoid the need that each area knows the global model and, last, but not least, the choice of abstraction level of the models.

The following example of the 12-bus meshed test system, shown in _{i}

The general characteristics of the DCMPC have been previously reported in [

However, a simple PV approximation [_{k}_{i}_{i} denote a candidate control sequence of LTC tap position changes and
_{i}^{N}^{H}^{−}^{N}^{+1} the corresponding finite set of all admissible control sequences, the size of
_{i}_{i}^{N}_{k}_{+}_{N}_{k}_{+}_{H}

The set of possible sequences of control values grows exponentially with the size of the network. Dealing with a very large-scale multi-area power system, such as the European interconnected network, some heuristics based on physical intuition would be needed in order to reduce the number of possible input sequences to be considered as possible MPC control, thus eliminating those that the designer knows in advance will almost certainly not be optimal. This paper serves as a proof-of-concept, and illustrates that for reasonably sized networks, as considered in this paper, the DCMPC is feasible for being applied in real time.

We decompose the overall power system into _{i}_{i}

This decomposition creates a very important “scalability” feature for the DCMPC scheme, which makes it suitable for dealing with large-scale multi-area power systems containing many distinct voltage control areas. This is thanks to the fact that each CA, in order to perform the local optimization, requires (in addition to the detailed local model) the QSS reduced-order models only of its few immediate neighboring areas (as well as limited information exchange only with them) and replaces the remaining distant areas by much simpler PV models without any information exchange. Thus, no matter how large the power system is, the computational complexity of the local calculations at each CA remains the same. Note that the decomposition criterion described above is applied to a given/existing power system with components already being physically in place, and the design of the system in order to best allocate the LTCs is out of the scope of this paper.

In order to develop a sufficiently fast simulator to cope with the hybrid dynamic nature of the power systems' behavior, we adopted a hybrid system model to formulate the long-term voltage control problem to be controlled by MPC-based controllers [^{®} [

An HA is a dynamical system describing the time evolution of a system involving the interaction of both continuous and discrete state variables. For example, the HA of _{1}, _{0}) ∈ _{q}_{1} (_{0}) = 0 is followed, as long as _{1}). As soon as the guard condition, Δ ∈ _{1}, _{2}), is fulfilled, the event _{1}, _{2}) ∈ _{2}, _{q}_{2} (_{2}), and so on. Note that

We focus on the long-term voltage instability in the time scale of 1 s to several minutes after a major disturbance. The long-term dynamics is typically driven by LTCs trying to locally restore the distribution side voltages of the connected buses and, hence, the corresponding voltage-dependent active/reactive powers of the dynamical recovery loads, while the OXL of synchronous generators may be activated, restricting their maximum reactive power generation capability [

blocking: fixing tap positions at their current positions;

locking: moving to a specific tap position and then blocking at this tap position;

reversing: changing the control logic to control the transmission side voltage instead of the distribution side [

voltage setpoint reduction: lowering the reference voltage.

The common drawback of all these rule-based logics is that they all first require identifying on which LTCs to act. This identification is not a trivial task, as it can be realized in many different ways. Furthermore, blocking and locking logics “kill” the tap-changing functionality of LTCs (by simply changing them to fixed transformers), ignoring all the voltage support contribution they can provide for voltage control. Moreover, on the other hand, these local heuristic rules may not suffice to face all possible scenarios in a large and complex power system. Thus, there is an essential need for a truly automatic model-based anticipation and coordination of LTCs' operation. To this end, the DCMPC approach, presented in this article and in our earlier work [

Given the slow control action for the long-term voltage control of interest in this article, the PMUupdates every 1 s would be still satisfactory for the practical implementation of the DCMPC scheme for collecting the local area measurements. For example, the (initial) tap positions of the LTCs and states of OXLs located within the area can be provided (say, every 1 s) by a local PMU. This local information collection might be even done via SCADA/EMSsystem in a lower time resolution.

Each CA, besides collecting the local information, is assumed to communicate, every _{c}_{c}

_{1} and _{2} (voltage/current or active/reactive powers over the tie-lines).

Considering Area 1 as an example and assuming that the Jacobian of _{1} with respect to
_{1}, at time _{k}

_{2}) on the control action of Area 1 (_{1}). Similarly, for _{2}:

Substituting _{2} can affect _{1}.

In the current implementation of the DCMPC approach in this article, we have taken this effect into account by providing to CA_{1} (corresponding to Area 1) the optimal control sequence of CA_{2} (corresponding to Area 2),
_{2} has planned to do in the interval [_{k−1},…, _{k+H−1}].

This seems to be an effective signal to be communicated, as the simulation results show that the uncoordinated local control actions can lead to voltage collapse. The exchange of the information of the planned control sequences allows each CA to “actively/directly” take its own coordinated local control actions, knowing what the neighbors are planning to do.

Another possible signal to be exchanged, instead of exchanging the planned control sequences, might be the exchange of the planned output trajectories of interacting variables directly (e.g., the expected values of future voltages over the tie-lines),

However, this seems to be much too complex, as the neighbors will “passively/indirectly” coordinate. Each CA is then expected to try to take into account the voltage profile of its neighbors by taking some local control actions and without knowing what the neighbors themselves will do in the future in order to correct their (potential) voltage violation. This might mislead the neighbors and stop them from reacting. This, in turn, can at least lead to a slower correction of a low voltage condition, assuming that the potential misunderstandings among neighbors can be resolved after some iteration of information exchange.

One should clearly distinguish between the local control models and the physical model of the overall system. There exist several local discrete time control models, one for each area, in order to enable the local MPC of each CA to perform its iterative finite-horizon optimization at each discrete time instant. These local control models obviously differ from one CA to another. However, on top of this, there also exists one continuous model for the overall physical power system (with information available from all areas) in order to simulate the dynamic response of the physical power system after applying the control actions calculated by the local MPCs. In other words, the DCMPC requires all local MPCs to concurrently calculate the local control actions for each time instant, say _{k}_{k}_{+1}. At _{k}_{+1}, all the state variables will be collected by performing new (noisy) measurements in the real physical power system, which will indeed be used as initial conditions for solving the local optimization problems by the local MPCs at _{k}_{+1}. This procedure is identically repeated for all following time instants. Thus, the CPU time required to complete a simulation experiment consists of two distinct terms: the time that the local MPCs take to calculate their optimal control actions at each discrete time instant, plus the time needed to simulate the physical system after applying these calculated control actions in order to obtain the state variables at the end of that control interval. In order to determine whether or not the DCMPC algorithm is suitable for real-time applications, what matters is the time needed for computing the local control actions for each CA (and not for all CAs) and, obviously, not for the simulation of the physical system. The DCMPC requires each CA to perform a number of simulation runs,

For the Nordic32 test system, the slowest agent-wise simulator time for solving the local optimization problem, using a variable step size solver, when running on a Windows PC with 3.154 GHz Intel Core 2 Duo CPU and 4 GB of RAM, was less than 3 s. Taking into account that each CA makes decisions at every 10 s, this ensures that the approach can meet the requirement for on-line voltage control.

Regarding why simulation stops and cannot proceed at the time of voltage collapse, first of all, due to step size reduction limitations, the simulation stops when the trajectory of the system variables changes too fast, as occurs during a voltage collapse. Moreover, the solver may not be able to find a solution to algebraic equations at the time of voltage collapse.

Recall that the hybrid nonlinear dynamic behavior of power systems involves an intrinsic strong coupling between continuous dynamics and discrete events. This occurs, for example, during voltage collapse phenomena, when several discrete devices (either discrete controllers, like LTCs or capacitor banks, or thresholds influencing continuous dynamics, like OXL) switch on and off as a reaction to local measurements of currents and voltages that are influenced by local and global continuous dynamics of the network.

To deal with this, as explained in Section 3.2., we have intentionally employed a hybrid modeling framework to formulate the long-term voltage coordination problem in Modelica (an object-oriented equation-based language to model complex physical systems), using Dymola (a tool for modeling and simulation of integrated and complex systems). Our hybrid Modelica models are described by a set of DAEs and discrete events. This hybrid nature leads to a mixed continuous/discrete system of equations at event instants that have to be solved by appropriate algorithms. Furthermore, different solvers may be used to determine the next simulation time (current simulation time plus step size) at which the set of DAEs, represented by the mathematical model of the system, should be solved. The difference between the true analytical solution and the computed approximated solution by numerical integration is defined as the local error. Generally, decreasing the step size increases accuracy, as well as simulation time, and

We have employed the variable-step solver of Dymola, which dynamically adjusts the step size during the simulation depending on the model dynamics and provides error control. Thus, by using local error, the solver reduces the step size to increase accuracy when a model's states are changing rapidly and increases the step size to avoid taking unnecessary steps when the model's states are changing slowly. The general effect will be that, for a given level of accuracy, the total number of steps will be reduced and, thus, overall simulation time, required to maintain a specified level of accuracy for models with rapidly changing or piecewise continuous states, will also be significantly reduced.

This indeed explains the behavior of the system when simulation stops and cannot proceed. When fast changes in voltage occur during collapse (which is the ultimate result of the activation of some OXLs, LTCs and their interactions), the solver tries to reduce the step size in order to maintain the required accuracy. However, since fast changes are so big that the required accuracy cannot be maintained, the solver fails to find a set of solutions and expectedly stops.

It is of interest in this article to find out what is the contribution of combining anticipation and coordination to improved performance of the DCMPC. The authors of [

Note that the control objective of each CA is two-fold. The main objective of CA_{i}_{i}_{i}_{i}

In comparison to the decentralized deadband control, only LTC _{2} reaches its maximum tap limit, while three other LTCs, namely _{1}, _{3} and _{4}, have not moved toward reaching their limits, thanks to the local MPC's anticipation. Furthermore, at _{2} successfully anticipates moving towards activation of OXL for _{4} fails at preventing the activation of OXL over

_{2}, still reaches its limit at _{13}, of final saturation.

In the simulations presented so far for the Nordic32 test system, the reduced-order QSS models for the immediate neighboring areas, as well as the PV models for the distant areas are assumed to be correctly given for each local area. It is of interest to perform additional experiments to assess the robustness of the DCMPC against modeling errors and to test the impact of the unmodeled dynamics of the neighbors. To this end, the consumed active power of the LTC-controlled load at Bus 4043 in Central_{6}, the most critical area with the most immediate neighbors containing three LTCs represented by a single equivalent one, is increased by 10% at _{1}, Central_{2}, Central_{4} and Central_{5}) for anticipating the power flows along the transmission lines connecting them to Central_{6} are not updated at _{6} at

The bus voltages are quite similar to (but lower than) those of the base disturbance in Case 1, all stabilized within the limits in the long term. However, in response to initiating (at _{6}, the local LTC, _{4} reacts to the nearby load change by taking a downward tap move at _{2} changes its (original base disturbance) LTC moves by freezing its upward moves at _{6}, Central_{4} and Central_{2}.

This article and [

A (potential) improvement for realizing practical applications might be achieved by the asynchronous implementation of the DCMPC. This may possibly provide an even more realistic operation of CAs by allowing them (in general) to update their control actions whenever they want to. In this way, the local CAs can base their decisions on more recent control plans from their neighbors than the previous synchronous time instant. However, due to the mechanical time delay of LTCs, the CAs would then still use the same control interval of 10 s, but could start reacting at different times within each 10-s interval.

It is also important to mention that the multi-area power systems are often heterogeneous networks stretching out over many independent TSOs (and their corresponding CAs) in different countries. Thus, the possible non-cooperative behavior of some CAs (or even preventing a malicious operator from actually damaging the overall system) must be rewarded, enforcing cooperation in practice. However, we do not deal with this issue, assuming that the TSOs adhere to their contract, which forces them to cooperate for the proposed DCMPC scheme.

In this article, at most, one LTC (except area Central_{6} containing three LTCs) is considered for each area. A control horizon of 30 s (

In the current implementation, each CA knows a detailed local model of its own area, as well as reduced-order QSS models of its immediate neighboring areas, assuming simpler equivalent PV model for the distant areas. Future works can be devoted to devising abstraction methods to dynamically represent a more abstract model of the neighboring areas. The abstract models may not necessarily have the same degree of accuracy and details compared to that of local areas.

This article does not consider the power-flow constraints of interconnecting tie-lines. These constraint are due to the thermal limits of tie-lines, which can, in turn, limit the maximum cross-border power interchange among areas. This additional constraint should also be taken into account in the future implementations of the DCMPC.

The DCMPC coordination scheme may also be applied as a tool for designing coordinated controllers for microgrids (

The proposed DCMPC approach is a novel communication-based control architecture for LTCs in order to improve the coordination among the corresponding CAs. This coordinating feedback controller acts in the time scale of 10 s and, thus, is suitable for mitigating the long-term voltage decays. Simulation results presented in this paper and [

The research was carried out in the frame of the Inter-university Attraction Poles program IAP-VII-02, funded by the Belgian Government.

The authors declare no conflict of interest.

control agent

model predictive control

transmission system operator

quasi steady-state

distributed communication-based model predictive control

load tap changing transformer

over excitation limiter

hybrid automaton

differential-algebraic equation

The model predictive control (MPC) concept.

A system controlled by MPC.

On-line diagram of a 12-bus power system.

Uncoordinated load tap changing transformer (LTC) moves and the corresponding bus voltages.

A simple hybrid automaton (HA).

A simple two-area interacting power system.

The Nordic32 partitioned interconnected test system.

Case 1 with the decentralized MPC approach: (

Case 1 with the decentralized MPC approach: (_{t}

Case 2 with the decentralized MPC approach: (

Case 2 with the decentralized MPC approach: (_{t}

Case 1 with 10% increase in the load at Bus 4043: (